Finally, we consider a toy model such that the pressure vanishes as \(r^{-n}\)  with \(n>2\) . There are then two families of trajectories describing wormholes. For one of them, the trajectories end at a singularity at finite \(r\)  and time. For the other one, the trajectories asymptotically tend to the Minkowski equilibrium point far from the throat. The physical result here, is that this flatness occur for both sides of the wormhole. This is thus a counterexample to a theorem claiming that a static wormhole sustained by a perfect fluid could not be flat on both sides of its throat. This is interesting since it has not to be glued to a vacuum exterior spacetime to be physically acceptable[1]}, [2]}.
